Deconvolution technique employing hermite functions

ABSTRACT

A procedure generates deconvolution algorithms by first solving a general convolution integral exactly. Results are transformed, yielding a linear relationship between actual (undistorted) and captured (distorted) data. Hermite functions and the Fourier-Hermite series represent the two data classes. It circumvents the need for solving incompatible systems of linear equations derived from “numerically discretizing” convolution integrals, i.e., the convolution integral is not evaluated. It is executed by exploiting a mathematical coincidence that the most common Point Spread Function (PSF) used to characterize a device is a Gaussian function that is also a Fourier-Hermite function of zero order. By expanding the undistorted data in a Fourier-Hermite series, the convolution integral becomes analytically integrable. It also avoids an inherent problem of dividing by decimal “noisy data” values in conventional “combined deconvolution” in that division is by a function of the PSF parameters yielding divisors generally greater than one.

STATEMENT OF GOVERNMENT INTEREST

[0001] Under paragraph 1(a) of Executive Order 10096, the conditionsunder which this invention was made entitle the Government of the UnitedStates, as represented by the Secretary of the Army, to the entireright, title and interest therein of any patent granted thereon by theUnited States. This and related patents are available for licensing toqualified licensees. Please contact John Griffin at 703 428-6265 orPhillip Stewart at 601 634-4113.

FIELD OF THE INVENTION

[0002] The present invention relates generally to mathematical processesfor removing distortion from raw data that presents in a Gaussiandistribution. In particular, it describes an efficient improvedde-convolution procedure for data processing adapted according tospecific needs of the user. It applies to all such measurementsincluding optical and electro-optical measurements.

BACKGROUND

[0003] Generally speaking, a measurement or imaging system introduces“distortion” into the “measurement” or image that is undesirable, i.e.,it distorts the desired information. This occurs in all measuringsystems. This undesirable distortion may be from a number of sourcesincluding the characteristics of the measuring system itself, theenvironment in which the measurable or image is embedded, theenvironment between the detector(s) of the measurement system and theitem to be measured or imaged, the experience of the operator of themeasuring system, etc. The resulting agglomeration of undesirabledistortions that have been “added” to the actual data results from a“convolution” of one or more of these sources of undesirable informationwith the desired (actual or undistorted) data by the measuring orimaging system.

[0004] The process for getting actual (undistorted) data from measured(distorted) data involves implementing a “deconvolution” that attemptsto reduce or eliminate the undesirable distortion. To do this,deconvolution techniques and deconvolution filters have been developed,some examples of which are represented in the following U.S. patents andpublished applications.

[0005] A number of U.S. patents involve some form of deconvolutiontechnique or filter for which the methods of the present invention mayprovide an improvement. These include the following examples.

[0006] U.S. Pat. No. 5,185,805, Tuned Deconvolution Digital Filter forElimination of Loudspeaker Output Blurring, to Chiang, Feb. 9, 1993, isincorporated herein by reference. The response of a speaker to a knownanalog signal is sampled at or above the Nyquist rate and the result isused to construct a linear, time invariant deconvolution filter thatcompacts the blurred output signal back into its original (unblurred)form. The frequency response is then improved by a fine-tuning processusing Lagrange's Method of Multipliers.

[0007] U.S. Pat. No. 5,400,265, Procedure for Enhancing Resolution ofSpectral Information, to Kauppinen, Mar. 21, 1995, is incorporatedherein by reference. The Fourier Self-Deconvolution (FSD) method is usedto enhance resolution of spectral data. The Maximum Entropy Method (MEM)is then used to predict filter error coefficients. Using data from theFSD and the MEM and the Linear Prediction (LP) method, data arepredicted beyond the area provided by the FSD to maximize spectral linenarrowing with minimal distortion.

[0008] U.S. Pat. No. 5,400,299, Seismic Vibrator SignatureDeconvolution, to Trantham, Mar. 21, 1995, is incorporated herein byreference. A deterministic signature deconvolution of the datacompresses the impulse response of the data resulting in sharper seismicimages than possible using cross correlation techniques.

[0009] U.S. Pat. No. 5,748,491, Deconvolution Method for the Analysis ofData Resulting from Analytical Separation Processes, to Allison et al.,May 5, 1998, is incorporated herein by reference. A signal representingmultiple partially separated sample zones is measured, a Point SpreadFunction (PSF) determined, and a Fourier transform of the dataperformed. A noise component of the signal is determined and a value fora resulting signal is determined using a specific filter. The inverseFourier Transform is then taken to get the desired result. Although aGaussian PSF is preferred for use in the process, a method is providedfor surveying a number of PSF possibilities serially.

[0010] U.S. Pat. No. 5,761,345, Method of Discrete Orthogonal BasisRestoration, to Moody, Jun. 2, 1998, is incorporated herein byreference. The method applies to linear data and involves estimating asignal-to-noise ratio (SNR) for a restored signal or image. A set oforthogonal basis functions is then selected to provide a stable inversesolution based on the estimated SNR. Time or spatially varyingdistortions in the restored system are removed and an appropriateinverse solution vector obtained.

[0011] U.S. Pat. No. 5,862,269, Apparatus and Method for RapidlyConvergent Parallel Processed Deconvolution, to Cohen et al., Jan. 19,1999, is incorporated herein by reference. Provided is a stand-aloneimage processing system and method for deconvolution or an initiator forrapid convergence of conventional deconvolution methods. The methodimproves processing speed by relaxing the sequential requirement of theCLEAN method. The number of iterations is reduced by fractional removalof noise for multiple image features during the processing of a singlesubtractive iteration.

[0012] U.S. patent application publication no. 2002/0156821 Al, SignalProcessing Using the Self-Deconvolving Data Reconstruction Algorithm, byCaron, Oct. 24, 2002, is incorporated herein by reference. A filterfunction is extracted from degraded data by mathematical operationsapplying a power law and smoothing function in frequency space. Thefilter function is used to restore the degraded content through anydeconvolution algorithm without prior knowledge of the detection system,i.e., blind deconvolution.

[0013] U.S. patent application publication no. 2002/0136133 Al,Superresolution in Periodic Data Storage Media, by Kraemer et al., Sep.26, 2002, is incorporated herein by reference. A method termedMatrix-Method Deconvolution (MMD) compensates for the effects of anoptical addressing system's PSF. Prior knowledge of the PSF andinter-memory center spacing is used to acquire binary data storedphysically in a periodic storage medium.

[0014] Optical and electro-optical devices cause intrinsic distortion ofthe image or spectral characteristic they sense, measure or record. Thisdistortion is separate from any measurement noise or artifact noise thatmay be present. The mathematical relationship between the undistortedand “intrinsically distorted” images or spectra is described by anequation incorporating a convolution integral. Blass, W. E. and G. W.Halsey, Deconvolution of Absorption Spectra, Academic Press, New York,1981. Jansson, P. A., Ed., Deconvolution of Images and Spectra, AcademicPress, New York, 1997.

[0015] The measured (distorted) image equals the convolution integral ofthe product of the actual (undistorted) image and the point spreadfunction (PSF) that stochastically characterizes the device. (Consider avery small point of light. If the visual system had perfect optics theimage of this point on a lens would be identical to the original pointof light. Referring to FIG. 1, if the relative intensity of this pointof light were plotted as a function of distance on the lens, such a plotwould look like the dashed vertical line. However, the optics are notperfect so the relative intensity of the point of light may bedistributed across the lens as shown by the curved line. This curve isthe PSF.)

[0016] Conventionally deriving an undistorted image or spectra from anintrinsically distorted one may be attempted initially by a stand-aloneprocess termed classical deconvolution. Unlike the convolution integral,deconvolution is not a well-defined mathematical operation, i.e., onlythe desired goal is well defined.

[0017] Because of the disadvantages of the deconvolution process, somehave offered solutions to completely avoid it. See, for example, U.S.Pat. No. 3,934,485, Method and Apparatus for Pulse Echo Imaging, toBeretsky et al., Jan. 27, 1976.

[0018] Some authors use the term “deconvolution” for what is actually acombination of classical (intrinsic) deconvolution and one of severalspecial iteration techniques. Blass (1981). Jansson (1997). Mou-yan, Z.et al., On the Computational Model of a Kind of Deconvolution Problem,IEEE Trans. Image Process, 4, No. 10, pp. 1464-1476, October 1995. Thisis a long-standing problem in deconvolution research, i.e., extractingan undistorted image from solutions to incompatible systems of linearequations derived from “discretizing” (not merely digitizing)convolution integrals. This discretizing yields a system of linearequations that relate the actual (undistorted) and captured (distorted)images in deconvolution. Twomey, S., Introduction to the Mathematics ofInversion in Remote Sensing and Indirect Measurements, Elsevier Sci.Pub., pp. 33-35, 1977.

[0019] Many classical deconvolution techniques, when used alone, producelarge unacceptable errors. To reduce these errors, the aforementionedlinear iteration techniques are used with intrinsic deconvolution. Usedtogether, the combination, after many iterations, may produce acceptableerror levels. Use of these combinations also may increasesignal-to-noise ratios (SNRs). Blass (1981). Jansson (1997). Mou-yan etal. (1995).

[0020] A recent patent details use of the Hermite Functions to improveperformance of a sonar system. U.S. Pat. No. 6,466,515 B1,Power-Efficient Sonar System Employing a Waveform and Processing Methodfor Improved Range Resolution at High Doppler Sensitivity, to Alsup etal., Oct. 15, 2002, is incorporated herein by reference. A sonar systemincorporates a comb-like waveform by modulating its lines according to aset of Hermite functions defining a Hermite Function Space (HFS).Doppler sensitivity akin to CW systems is realized by applying to theHFS signals, the deconvolution method of the '515 patent. AlthoughHermite Functions are used, they are not used to simplify thedeconvolution of received data.

[0021] Thus, although a number of “combined” deconvolution techniquesare known, difficulties arise in applying many of them. The presentinvention provides a solution that overcomes such difficulties.

SUMMARY

[0022] A unique deconvolution procedure adapts to generate uniquedeconvolution algorithms for each different requirement. This isaccomplished by first solving the general convolution equation exactly,in closed analytical form. Mou-Yan, Z. et al., New Algorithms ofTwo-Dimensional Blind Deconvolution, Opt. Eng., No. 10, pp. 2945-2956,October 1995. Next, the results are transformed, leading to afundamentally new linear relationship between the actual (undistorted)and the captured (distorted) images.

[0023] To do this, a preferred embodiment of the present invention usesthe Hermite functions and the Fourier-Hermite series to represent thetwo classes (distorted and undistorted) of images. Using this procedureleads to the exact solution of the convolution integral.

[0024] A preferred embodiment of the present invention provides a wayaround the need for solving incompatible systems of linear equationsderived from “discretizing” convolution integrals. No numericalevaluation of the convolution integral is required in the presentinvention.

[0025] A preferred embodiment of the present invention is executed byexploiting a remarkable mathematical coincidence. The mathematicalcoincidence is that the most common PSF used to characterize a device isa Gaussian function (see FIG. 1). The Gaussian function is also aFourier-Hermite function of zero order.

[0026] Exploitation of this fact permits exact analytic evaluation ofthe convolution integral for most optical imaging systems and somenon-optical systems. The mathematics is processed in exact closed form,followed by analytic deconvolution. Further, the present inventionaddresses an additional concern. When employing conventional “combineddeconvolution,” image distortion may also result because of thenumerical division of very noisy data represented by PSF “points” havingvalues less than one. Division in the present invention is by a newfunction of the PSF parameters yielding values generally greater thanone.

[0027] If desired, the procedure may be used with any of the iterativetechniques to further refine the image. Finally, if used with aniterative technique, fewer iterations will be needed because of theinherently more accurate starting point provided by the procedure.

[0028] Specifically provided is a process for deconvolving datacollected by a device the point spread function response of whichfollows a Gaussian distribution. The process accurately estimates actualparameters derivable from this “captured” data. It comprises:

[0029] forming a mathematical relationship having a first mathematicalequivalent of the captured data on one side of an equality and a secondmathematical equivalent of actual parameters on the other side of theequality;

[0030] selecting an order, m, of a Hermite function for modifying thismathematical relationship;

[0031] modifying the mathematical relationship to form Hermite functionstherein to permit identification of like items, each having acoefficient, on each side of the equality, the coefficients associatedwith the actual parameters being unknowns;

[0032] developing a set of linear equations from this mathematicalrelationship that relate to the coefficients of the like items; and

[0033] solving this set of linear equations for the unknown coefficientsto produce an exact solution to the convolution; and

[0034] deconvolving this set of linear equations to determine theunknown coefficients.

[0035] The process may also initiate any conventional iterativedeconvolution techniques to further refine the data, such that feweriterations will be needed because of provision of an accurate startingpoint.

[0036] Specifically, a preferred embodiment of the present invention isa process for deconvolving output from detectors whose point spreadfunction follows a Gaussian distribution. The process accuratelyestimates environments derivable from the detectors' output. The processcomprises:

[0037] forming a mathematical relationship having the mathematicalequivalent of the output on one side of an equality and the mathematicalequivalent of the environments on the other side;

[0038] selecting an order, m, of a Hermite function for modifying thisequality, such that m also determines the number of terms for therepresentation of the environments;

[0039] modifying the mathematical relationship to form Hermite functionstherein, such that forming the Hermite functions permits identificationof like items, each having a coefficient, on each side of the equality;

[0040] developing a set of linear equations from the equality thatrelate to the coefficient of each of the like items; and

[0041] solving the set of linear equations for the unknown coefficientsof the like items to produce an exact solution to the convolution, inturn, permitting deconvolution using simple linear relationships.

[0042] Also, this embodiment may be used to initiate any conventionaliterative nonlinear deconvolution techniques to further refine theoutput such that fewer iterations are needed because of the providedaccurate starting point.

[0043] In yet another preferred embodiment of the present invention isprovided a process yielding accurate representations of actual imagedata by deconvolving image data collected by optical detectors whosepoint spread function response follows a Gaussian distribution. Theprocess comprises:

[0044] establishing a mathematical relationship as a general opticalconvolution integral equating the actual image data to the collectedimage data;

[0045] selecting a Fourier-Hermite function;

[0046] employing a generating function for Hermite polynomials forexpanding a Fourier-Hermite series of the Fourier-Hermite function toestablish a linear mathematical relationship between the actual and thecollected image data;

[0047] selecting an order, m, of the Fourier-Hermite polynomial suchthat m also determines the number of terms for the representation of theimages;

[0048] expanding the actual image data in a Fourier-Hermite form withunknown (to-be-determined) coefficients by employing a series of specialtransformations to convert the side of the mathematical relationshiprepresenting the actual image to a Fourier-Hermite series;

[0049] expanding the collected image data in a Fourier-Hermite form;

[0050] equating coefficients of like terms on each side of themathematical relationship to relate the coefficients of the actual andcollected images;

[0051] selecting a linear convolution algorithm from which a generalconvolutional equation is derived;

[0052] solving the resultant general convolution equation exactly, inclosed analytical form, as a linear convolution equation, such thatusing this process to evaluate the general convolution integral yields aform proportional to a Gaussian function times a power series that isdefined with a finite number of terms, m, and incorporates the unknowncoefficients of the actual image data as presented in a Hermitefunction, such that a solution in closed analytic form provides asatisfactory solution to the general convolution integral withoutapproximation; and

[0053] performing an analytic deconvolution of the resultant linearconvolution algorithm by employing classical deconvolution algorithmssolely, such that the analytic deconvolution yields a solution havingacceptable error levels.

[0054] As with other embodiments, this process may be used to initiateany conventional iterative nonlinear deconvolution techniques to furtherrefine the accurate representations and fewer iterations are neededbecause of the accurate starting point provided.

[0055] A preferred embodiment of the present invention for deconvolvingcaptured (distorted) optical, or other, measurements has the followingadvantages:

[0056] it completely circumvents the long-standing problem of having touse solutions from incompatible systems of linear equations derived fromdiscretizing convolution integrals;

[0057] it eliminates concern about discernible image distortionintroduced by the imaging process itself in addition to that from themeasuring device, e.g., a detector array;

[0058] it automatically provides the unique algorithm(s) needed for thespecific requirements of the user;

[0059] it solves both the convolution and the deconvolution algorithmsexactly through a system of related linear equations, not as an estimatebased on an iterative process;

[0060] it may be used with existing iterative processes, reducing thenumber of iterations required while further refining an image, ifrequired; and

[0061] it capitalizes on the unique characteristics of Hermitefunctions, enabling calculations in the frequency (wave number) domainas easily as in the time domain.

[0062] Applications in the imaging field include:

[0063] straightforward calculation of the actual (undistorted)measurement from the collection system's (distorted) measurement;

[0064] removal of distortions from image data; and

[0065] assessment of the accuracy of a given convolution.

[0066] Further advantages of the present invention will be apparent fromthe description below with reference to the accompanying drawings, inwhich like numbers indicate like elements.

BRIEF DESCRIPTION OF THE DRAWINGS

[0067]FIG. 1 is a representation of a point spread function (PSF)compared to a theoretically perfect representation of a single point oflight as may be captured by a lens.

[0068]FIG. 2 is a flow chart representing general steps used inimplementing a preferred embodiment of the present invention.

DETAILED DESCRIPTION

[0069] There is concern about some of the numerical division operationsthat are a part of deconvolution algorithms. Concerns center on thedivision of a value associated with a signal datum that is mostly noiseby the actual value of the PSF when the PSF is a very small percentageof its peak value, such as in the tails of a PSF represented by aGaussian distribution. Of course, the peak value of the PSF, bydefinition, is one. It can be seen that such operations, i.e., divisionby a number much smaller than one, will magnify the contribution of thenoisy datum. This produces significant distortion as a result ofprocessing alone. This is then combined with the physical distortionintroduced by the collection device itself.

[0070] The Hermite-function method, proposed as a preferred embodimentof the present invention, prevents, or at least significantly reducesthe chances of, these small values being used as a divisor. Thisalgorithm provides for dividing by a function of the parameters of thePSF, as opposed to the actual values of the PSF. Thus, proposed is afunction of the PSF, ƒ(PSF), independent of the point at which the PSFis being used. For most devices, this function is rarely much less thanone, leading to little chance of magnifying a noisy datum.

[0071] The actual (undistorted) measurement is expanded in aFourier-Hermite form with to-be-determined coefficients. This leads tothe need to evaluate convolution integrals of the form: $\begin{matrix}{Y_{m} = {\int_{- \infty}^{\infty}{^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{H_{m}(x)}{x}}}} & (1)\end{matrix}$

[0072] where:

[0073] PSF is represented by:${Pe}^{\frac{x^{2}}{2}},\quad {0 < P < 1},$

[0074] displaced PSF is represented by $^{\frac{{({z - x})}^{2}}{2}},$

[0075] the Hermite Function is represented by${^{\frac{- x^{2}}{2}}{H_{m}(x)}},$

[0076]  and

[0077] H_(m)(x) is the Hermite polynomial of order m.

[0078] By way of introduction to the development of the theory behindthe present invention, the “convolutional” relationship between the trueand distorted measurements (or images) is given by:

[0079] The recorded (distorted) image may be expressed as a convolutionintegral of the form:

∫[PSF(x−y)][TRUEIMAGE(y)]dy  (2)

[0080] Suppose the convolutional equation relating the distorted andundistorted images could be solved exactly. Further, suppose the twosides of the equation can be manipulated to be of the same form. Thus,equating coefficients of like terms directly relates the parameters ofthe two images (distorted and actual), ultimately determining thedesired algorithm.

[0081] An expression for an actual (undistorted) image, I(x), in termsof a Hermite function is: $\begin{matrix}{{I(x)} = {^{- \frac{y^{2}}{2}}{\sum\limits_{m = 0}^{\infty}{I_{m}{H_{m}(y)}}}}} & (3)\end{matrix}$

[0082] where H_(m)(y) is the Hermite polynomial of order, m; and$\begin{matrix}{y = {\alpha^{\frac{1}{2}}x}} & (4)\end{matrix}$

[0083] where:

[0084] x the spatial variable and

[0085] α=an arbitrary parameter

[0086] Pauling, L. and E. Bright Wilson, Introduction to QuantumMechanics, p. 80, McGraw-Hill, 1935. The Pauling reference includes anormalization factor that is a function of α. Since the factor is aconstant, it has been absorbed in I_(m).

[0087] The general convolutional relationship between the actual(undistorted) and captured (distorted) measurements (images) is givenby:

D(z)=∫p(z−x)I(x)dx  (5)

[0088] where:

[0089] D(z)=captured measurement;

[0090] I(x)=actual value of the measurement;

[0091] p(x)=point spread function (PSF) characterizing the measuringdevice, such as an optical or electro-optical imaging system

[0092] In most cases, p(x) may be described by: $\begin{matrix}{{p(x)} = {P\quad ^{- \frac{{({x - d})}^{2}}{2a}}}} & (6)\end{matrix}$

[0093] where:

[0094] P is (2πb)^(1/2), the multiplicative “amplitude” of the Gaussian;

[0095] a is a constant related to a measure of the “width” of theGaussian;

[0096] b is a constant that is an inverse measure of the “amplitude” ofthe Gaussian; and

[0097] d is the displacement of Gaussian peak from the origin, normallyzero

[0098] Thus, the resulting form of P is in terms of a and b (d normallyzero). When a=b, the integral of p(x) over all x is unity. P can neverbe greater than unity and for simplicity it is chosen as unity.

[0099] When the PSF can be modeled by a Gaussian curve written as$\begin{matrix}{{p(x)} = {P\quad ^{\frac{- x^{2}}{2a}}}} & (7)\end{matrix}$

[0100] where:

[0101] a=a constant,

[0102] b=a constant

[0103] Expanding the unknown undistorted measurement (actual image),I(x), and the measured (distorted image), D(R), in Fourier-Hermiterepresentations: $\begin{matrix}{{I(x)} = {^{- \frac{\alpha \quad x^{2}}{2}}{\sum\limits_{m = 0}^{\infty}{I_{m}{H_{m}\left( {x\quad \alpha^{\frac{1}{2}}} \right)}}}}} & (8) \\{and} & \quad \\{{D(R)} = {\sum\limits_{m = 0}^{\infty}{D_{m}^{- \frac{R^{2}}{2}}{H_{m}(R)}}}} & (9)\end{matrix}$

[0104] I_(m) represents unknown coefficients that when solved forrepresent the solution of the deconvolution problem. D_(m) representsknown coefficients needed to represent D(R). Also, α is and arbitraryparameter that is later constrained by the requirement of physical“realizability” of the description.

[0105] D(R) and I(x) are related through the definitive equationcontaining the convolutional integral $\begin{matrix}{{D(z)} = {\int_{- \infty}^{\infty}{{p\left( {z - x} \right)}{I(x)}{x}}}} & (10)\end{matrix}$

[0106] Integrating the right hand side (RHS) of Eqn. (10), a polynomial,I_(m)(R)^(m), results. This polynomial must be transformed into theHermite Polynomial form, J_(m)H_(m)(R), where J_(m) is related to I_(m)by a linear transformation the details of which depend on the order ofthe polynomial, m.

[0107] The results produce identical structure on both sides of theequation, enabling the coefficients of like terms to be equated. A setof linear equations relating parameters of the distorted and undistortedimages may now be written. Writing: $\begin{matrix}{C = \left( \frac{a}{2b} \right)^{\frac{1}{2}}} & (11)\end{matrix}$

[0108] yields a system of linear equations:

D _(m) =CJ _(m)  (12) $C = \left( \frac{a}{2b} \right)^{\frac{1}{2}}$

[0109] that may be used to solve the convolution problem.

[0110] Further, Eqn. (11) and (12) may be used to solve for J_(m) andhence, I_(m), yielding the solution to the deconvolution problem. Thecommon divisor, C, or a common multiplier, E, may be used, where E=1/C.Thus, $\begin{matrix}{E = \left( \frac{2b}{a} \right)^{\frac{1}{2}}} & (13)\end{matrix}$

[0111] so that

J _(m) =ED _(m)  (14)

[0112] Thus, the solution to the deconvolution problem can be expressedas $\begin{matrix}{D_{n} = {{\sum\limits_{m = 0}^{n}{C_{n}J_{m}}} = {C\quad {\sum\limits_{m = 0}^{n}J_{m}}}}} & (15)\end{matrix}$

[0113] A process leading to a relationship equivalent to the integral ofEqn. (1) but able to be easily evaluated establishes a preferredembodiment of the present invention. The artifice of using a generatingfunction for Hermite polynomials enables this.

[0114] One generating function for Hermite polynomials is:$\begin{matrix}{{T\left( {x,t} \right)} = {^{{- t^{2}} + {2{tx}}} = {\sum{\left\lbrack \frac{H_{m}(x)}{m!} \right\rbrack \quad t^{m}}}}} & (16)\end{matrix}$

[0115] where t is a dummy variable.

[0116] A usable integral may be created using the generating function asfollows: $\begin{matrix}{\Pi = {\int{{T\left( {x,t} \right)}\quad ^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{x}}}} & (17)\end{matrix}$

[0117] Substituting the exponential definition in Eqn. (1) into Eqn.(17): $\begin{matrix}{\Pi = {\int{^{{- t^{2}} + {2{tx}}}\quad ^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{x}}}} & (18)\end{matrix}$

[0118] or the alternative summation definition in Eqn. (16) into Eqn.(17): $\begin{matrix}{\Pi = {\int{^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{\sum{\left\lbrack \frac{H_{m}(x)}{m!} \right\rbrack t^{m}{x}}}}}} & (19)\end{matrix}$

[0119] or: $\begin{matrix}{\Pi = {\frac{t^{m}}{m!}{\sum{\int\quad {^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{H_{m}(x)}{x}}}}}} & (20)\end{matrix}$

[0120] Note that the integral in Eqn. (20) is Y_(m) of Eqn. (1).

[0121] Setting the right hand side of Eqn. (18) equal to the right handside of Eqn. (20) produces: $\begin{matrix}{{\int{e^{{- t^{2}} + {2{tx}}}^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{x}}} = {\frac{t^{m}}{m!}{\sum{\int\quad {^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{H_{m}(x)}{x}}}}}} & (21)\end{matrix}$

[0122] Noting that exponents add, collecting all exponents within theintegral on the right hand side of Eqn. (17), and setting this to equalE: $\begin{matrix}{E = {- \left\lbrack {\left( {t^{2} - {2{tx}}} \right) + \frac{\left( {z - x} \right)^{2}}{2} + \frac{x^{2}}{2}} \right\rbrack}} & (22) \\{{or}\text{:}} & \quad \\{E = {- \left\lbrack {t^{2} - {2{tx}} + \frac{\left( {z^{2} - {2{zx}} + x^{2}} \right)}{2} + \frac{x^{2}}{2}} \right\rbrack}} & (23) \\{{or}\text{:}} & \quad \\{E = {- \left( {t^{2} - {2{tx}} + \frac{z^{2}}{2} - {zx} + \frac{x^{2}}{2} + \frac{x^{2}}{2}} \right)}} & (24) \\{{or}\text{:}} & \quad \\{E = {- \left\lbrack {t^{2} - {x\left( {{2t} + z} \right)} + \frac{z^{2}}{2} + x^{2}} \right\rbrack}} & (25)\end{matrix}$

[0123] Define:

z=2y  (26)

[0124] and rewrite Eqn. (25) using Eqn. (26):

E=−[t ²−2x(t+y)+2y ² +x ²]  (27)

[0125] Also define:

F=−[x−(y+t)]²  (28)

[0126] or:

F=−[x ²−2x(y+t)+(y+t)²]  (29)

[0127] or:

F=−[x ²−2x(y+t)+y ²+2yt+t ²]  (30)

[0128] Thus,

E−F=−[y ²−2yt]  (31)

[0129] Substituting Eqn. (27) into the left hand side of Eqn. (17) anddefining this as R:

R=∫e ^(−(F+y) ² ^(−2yt)) dx  (32)

[0130] or:

R=e ^(−(y) ² ^(−2yt)) ∫e ^(−F) dx  (33)

[0131] so that:

R=e ^(−(y) ² ^(−2yt)) ∫e ^(−[x−(y+t)]) ² dx  (34)

[0132] and, since y and t are constants, thus dx=dy=0, then R may beexpressed as:

R=e ^(−(y) ² ^(−2yt)) ∫e ^(−[x−(y+t)]) ² d[x−(y+t)]  (35)

[0133] Setting:

w=x−(y+t)  (36)

[0134] then:

R=e ^(−(y) ² ^(−2yt)) ∫e ^(−w) ² dw  (37)

[0135] but: $\begin{matrix}{{\int_{- \infty}^{\infty}{^{- w^{2}}{w}}} = \pi^{\frac{1}{2}}} & (38)\end{matrix}$

[0136] Thus, Eqn. (37) becomes: $\begin{matrix}{R = {\pi^{\frac{1}{2}}^{- {({y^{2} - {2{yt}}})}}}} & (39)\end{matrix}$

[0137] Substituting Eqn. (39) into Eqn. (20): $\begin{matrix}{{\sum{\frac{t^{m}}{m!}Y_{m}}} = {R = {\pi^{\frac{1}{2}}^{- {({y^{2} - {2{yt}}})}}}}} & (40)\end{matrix}$

[0138] A Taylor Expansion of e^(2yt) provides: $\begin{matrix}{^{2\quad {yt}} = {1 + \frac{2{yt}}{1!} + \frac{\left( {2{yt}} \right)^{2}}{2!} + \ldots + \frac{\left( {2{yt}} \right)^{m}}{m!} + \ldots}} & (41) \\{{or}\text{:}} & \quad \\{^{2\quad {yt}} = {\sum\frac{\left( {2{yt}} \right)^{m}}{m!}}} & (42)\end{matrix}$

[0139] Substituting Eqn. (42) into Eqn. (40) yields: $\begin{matrix}{{\sum{\frac{t^{m}}{m!}Y_{m}}} = {\pi^{\frac{1}{2}}^{- y^{2}}{\sum\frac{\left( {2{yt}} \right)^{m}}{m!}}}} & (43)\end{matrix}$

[0140] Equating coefficients of like terms: $\begin{matrix}{{\frac{t^{m}}{m!}Y_{m}} = {\pi^{\frac{1}{2}}^{- y^{2}}\quad \frac{\left( {2{yt}} \right)^{m}}{m!}}} & (44)\end{matrix}$

[0141] such that: $\begin{matrix}{Y_{m} = {\pi^{\frac{1}{2}}{^{- y^{2}}\left( {2y} \right)}^{m}}} & (45)\end{matrix}$

[0142] Substituting Eqn. (26) in Eqn. (45): $\begin{matrix}{Y_{m} = {\pi^{\frac{1}{2}}{^{- \frac{z^{2}}{4}}(z)}^{m}}} & (46)\end{matrix}$

[0143] Upon integration of Eqn. (5), a polynomial form, I_(m)(R)^(m),results and must be transformed back into the Hermite polynomial form,J_(m)H_(m)(R), where J_(m) is related to I_(m) by a lineartransformation which depends on the order, m, of the polynomial.

[0144] Eqn. (5) can be solved exactly. Further, the two sides of Eqn.(5) may be represented in the same form. Equating coefficients of liketerms enables direct relationship of the parameters of the twomeasurements (actual and captured), which then determines the requiredalgorithms.

[0145] The present invention makes use of the observation that mostoptical and electro-optical devices, as well as many other measurementsystems, are characterized by a Gaussian PSF. Only the constants in theGaussian vary from system to system. Note that the PSF appears withinthe convolution integral. Three aspects of the solution merge here.

[0146] First, Gaussian functions are part of a larger class of functionsand are Hermite functions of zero order. Second, the actual(undistorted) measurement or image usually may be represented by aFourier-Hermite series. Note also that the actual (undistorted)measurement or image is also defined within the convolution integral.Finally, the combination of the first two aspects allows theclosed-form, analytic evaluation of the nearly general convolutionintegral without approximation, with the exception that the series arefinite in length. In the above case, it is an optical convolutionintegral, but the process would work for any device the response ofwhich may be described with a Gaussian distribution. Thus, using thisapproach, the common method of “discretizing” is avoided and there is nolarge system of linear equations that may be incompatible and may leadto meaningless results.

[0147] A preferred embodiment of the present invention, i.e., anevaluation of the convolution integral, yields a form proportional to aGaussian function times a power series involving the unknown parametersof the actual (undistorted) measurement or image. Using a series ofspecial transformations, the side of the equation representing theactual (undistorted) image may be changed into a separateFourier-Hermite series.

[0148] Since the side of the equation representing the captured(distorted) measurement or image may be represented by a Fourier-Hermiteseries also, the two sides will have identical structure. Equatingcoefficients of like terms now directly relates the parameters of thetwo measurements or images (actual and captured), and permits selectionof the required algorithms. The only non-numeric constants in theresults are the parameters of the PSF that have been measured for theindividual device or device type. Developing this concept further bysubstituting Eqn. (6) for p(x) into Eqn. (5) yields: $\begin{matrix}{{D(z)} = {\int{^{- \frac{{({z - x})}^{2}}{2a}}{I(y)}{x}}}} & (47)\end{matrix}$

[0149] Substituting Eqns. (3) and (4) into Eqn. (47) yields:$\begin{matrix}{{D(z)} = {\int{^{- \frac{{({z - x})}^{2}}{2a}}^{- \frac{y^{2}}{2}}{\sum{I_{m}{H_{m}(y)}{x}}}}}} & (48) \\{or} & \quad \\{{D(z)} = {P{\sum{I_{m}{\int{^{- \frac{{({z - x})}^{2}}{2a}}{H_{m}(y)}^{- \frac{y^{2}}{2}}{x}}}}}}} & (49)\end{matrix}$

[0150] If the integral on the right side of Eqn. (49) are designatedY_(m), then: $\begin{matrix}{Y_{m} = {\int{^{- \frac{{({z - x})}^{2}}{2a}}^{- \frac{y^{2}}{2}}{H_{m}(y)}{x}}}} & (50)\end{matrix}$

[0151] Since Y_(m) represents a value for the m^(th) mode, the sumsubstituted in Eqn. (49) is:

D(z)=PΣI _(m) Y _(m)  (51)

[0152] To evaluate the integral, choose: $\begin{matrix}{^{- \frac{{(x)}^{2}}{2a}} = {^{- \frac{y^{2}}{2}} = ^{- \frac{x^{2}\alpha}{2}}}} & (52) \\{{Thus},} & \quad \\{\frac{x^{2}}{2a} = \frac{x^{2}\alpha}{2}} & (53) \\{or} & \quad \\{\alpha = \frac{1}{a}} & (54)\end{matrix}$

[0153] Substituting Eqns. (4) and (54) into Eqn. (3): $\begin{matrix}{{I(x)} = {^{- \frac{x^{2}}{2a}}{\sum{I_{m}{H_{m}(x)}a^{- \frac{1}{2}}}}}} & (55)\end{matrix}$

[0154] and the integral in Eqn. (50) becomes: $\begin{matrix}{Y_{m} = {\int{^{- \frac{{({z - x})}^{2}}{2a}}^{- \frac{x^{2}}{2a}}{H_{m}(x)}a^{- \frac{1}{2}}{x}}}} & (56)\end{matrix}$

[0155] Let: $\begin{matrix}{v = {a^{- \frac{1}{2}}x}} & (57)\end{matrix}$

[0156] so that $\begin{matrix}{{d\quad x} = {a^{\frac{1}{2}}d\quad v}} & (58)\end{matrix}$

[0157] and let $\begin{matrix}{t = {a^{- \frac{1}{2}}z}} & (59)\end{matrix}$

[0158] then Eqn. (56) may be re-written as: $\begin{matrix}{Y_{m} = {a^{\frac{1}{2}}{\int{^{- \frac{{({t - v})}^{2}}{2}}^{- \frac{v^{2}}{2}}{H_{m}(v)}{v}}}}} & (60)\end{matrix}$

[0159] Eqn. (60) is now of the form of the convolution integral wherea=1. The result obtained for the convolution integral is:$\begin{matrix}{Y_{m} = {\pi^{\frac{1}{2}}^{- \frac{t^{2}}{4}}t^{m}}} & (61)\end{matrix}$

[0160] and for a≠1 is: $\begin{matrix}{Y_{m} = {\left( {a\quad \pi} \right)^{\frac{1}{2}}^{- \frac{t^{2}}{4}}t^{m}}} & (62)\end{matrix}$

[0161] Substituting the definition of t from Eqn. (59) into Eqn. (62)yields: $\begin{matrix}{Y_{m} = {\left( {a\quad \pi} \right)^{\frac{1}{2}}{^{- \frac{z^{2}}{4a}}\left( {z\quad a^{\frac{1}{2}}} \right)}^{m}}} & (63)\end{matrix}$

[0162] To represent as a Fourier-Hermite series, let: $\begin{matrix}{\frac{R^{2}}{2} = \frac{z^{2}}{4a}} & (64) \\{or} & \quad \\{R = \frac{z}{\left( {2a} \right)^{\frac{1}{2}}}} & (65)\end{matrix}$

[0163] so that Eqn. (63) may be written: $\begin{matrix}{Y_{m} = {\left( {\pi \quad a} \right)^{\frac{1}{2}}2^{\frac{m}{2}}^{- \frac{R^{2}}{2}}R^{m}}} & (66)\end{matrix}$

[0164] Multiplying (πa)^(1/2) by P⁻¹ or (2πb)^(−1/2) yields:$\begin{matrix}{Y_{m} = {\left( \frac{a}{2b} \right)^{\frac{1}{2}}2^{\frac{m}{2}}^{- \frac{R^{2}}{2}}R^{m}}} & (67)\end{matrix}$

[0165] Inserting Eqn. (67) in Eqn. (51) produces: $\begin{matrix}{{D(z)} = {\left( \frac{a}{2b} \right)^{\frac{1}{2}}{\sum{{I_{m}(2)}^{\frac{m}{2}}{^{- \frac{R^{2}}{2}}(R)}^{m}}}}} & (68)\end{matrix}$

[0166] This expression now approaches the form of a Fourier-Hermiteseries because R^(m) can be expressed as a linear sum of weightedHermite polynomials. Rewriting Eqn. (68): $\begin{matrix}{{D(z)} = {\left( \frac{a}{2b} \right)^{\frac{1}{2}}{\sum{2^{\frac{m}{2}}^{- \frac{R^{2}}{2}}J_{m}{H_{m}(R)}}}}} & (69)\end{matrix}$

[0167] Note that the Fourier-Hermite series can represent an almostarbitrary function in the same sense that the common sin(x), cos(x)series of the Fourier Series does.

[0168] The Fourier-Hermite representation of D(R) has the general form:$\begin{matrix}{{D(R)} = {\sum\limits_{m = o}^{\infty}{D_{m}^{- \frac{R^{2}}{2}}{H_{m}(R)}}}} & (70)\end{matrix}$

[0169] where values of D_(m) are constants and the m^(th) value ofH_(m)(R) is represented by the m^(th) order Hermite polynomial. D_(m)values are determined from H_(m)(R) by: $\begin{matrix}{D_{m} = {\left( {{m!}2m\quad \pi} \right)^{- 1}{\int_{- \infty}^{\infty}{{D(R)}^{- \frac{R^{2}}{2}}{H_{m}(R)}{R}}}}} & (71)\end{matrix}$

[0170] In practice, a finite number of terms in the series of Eqn. (70)is used to represent D(R). Assuming that finite number to be M, Eqn.(70) may be re-written as: $\begin{matrix}{{D(R)} = {\sum\limits_{m = o}^{M}{D_{m}^{- \frac{R^{2}}{2}}{H_{m}(R)}}}} & (72)\end{matrix}$

[0171] The number of terms, M, selected here will also determine thenumber of terms for the representation of the actual (undistorted)measurement or image, I(x). The value chosen for M determines thespecific structure of the algorithm that the procedure produces. Thisprovides some flexibility for the user in meeting the possibly competingrequirements of computational efficiency and image “sharpness,” forexample.

[0172] Based on the “basic” convolution equation of Eqn. (5), substitutethe RHS of Eqn. (69) for the LHS of Eqn. (5) and the RHS of Eqn. (72)for the RHS of Eqn. (5), yielding: $\begin{matrix}{{^{- \frac{R^{2}}{2}}{\sum\limits_{m = o}^{M}{D_{m}{H_{m}(R)}}}} = {\left( \frac{a}{2b} \right)^{\frac{1}{2}}^{- \frac{R^{2}}{2}}{\sum\limits_{m = 0}^{M}{2^{\frac{m}{2}}J_{m}{H_{m}(R)}}}}} & (73)\end{matrix}$

[0173] Because both sides of Eqn. (73) now have identical structures,coefficients of like terms may be equated. A set of linear equationsrelating parameters of the captured (distorted) and actual (undistorted)images may be produced. Using the set of Eqns. (11) and (12) provides asystem of linear equations that can be solved for J_(m) and then I_(m),yielding the exact solution to the deconvolution. Using the commonmultiplier, E, and Eqn. (14), Eqn. (15) may be expressed alternativelyas: $\begin{matrix}{I_{m} = {{\sum\limits_{m = 0}^{M}{A_{m}J_{m}}} = {E{\sum\limits_{m = 0}^{M}{A_{m}D_{m}}}}}} & (74)\end{matrix}$

[0174] representing the solution of the deconvolution, where A_(m) isthe “structural” part of the relationship between I_(m) and J_(m). Notethat I_(m) represents the coefficients of the actual (undistorted)measurement or image and J_(m) represents a set of dummy variables. Itis desirable to be able to move back and forth between the two sets.

[0175] Refer to FIG. 2, a flow chart of a process yielding accuraterepresentations of actual measurement or image data by deconvolving datacollected by systems, such as optical detectors, whose point spreadfunction response follows a Gaussian distribution.

[0176] The process comprises:

[0177] establishing 201 a mathematical relationship as a generalconvolution integral equating the actual measurement or image data tothe collected measurement or image data;

[0178] selecting 202 a Fourier-Hermite function;

[0179] employing 203 a generating function for Hermite polynomials forexpanding a Fourier-Hermite series of the Fourier-Hermite function toestablish a linear mathematical relationship between the actual and thecollected measurement or image data;

[0180] selecting 204 an order, m, of the Fourier-Hermite polynomial suchthat m also determines the number of terms for the representation of themeasurement or images;

[0181] expanding 205 the actual measurement or image data in aFourier-Hermite form with unknown (to-be-determined) coefficients byemploying a series of special transformations to convert the side of themathematical relationship representing the actual measurement or imageto a Fourier-Hermite series;

[0182] expanding 206 the collected measurement or image data in aFourier-Hermite form;

[0183] equating 207 coefficients of like terms on each side of themathematical relationship to relate the coefficients of the actual andcollected measurement or images;

[0184] selecting 208 a linear convolution algorithm from which a generalconvolutional equation is derived;

[0185] solving 209 the resultant general convolution equation exactly,in closed analytical form, as a linear convolution equation, such thatusing this process to evaluate the general convolution integral yields aform proportional to a Gaussian function times a power series that isdefined with a finite number of terms, m, and incorporates the unknowncoefficients of the actual measurement or image data as presented in aHermite function, such that a solution in closed analytic form providesa satisfactory solution to the general convolution integral withoutapproximation; and

[0186] performing 210 an analytic deconvolution of the resultant linearconvolution algorithm by employing classical deconvolution algorithmssolely;

[0187] evaluating 211 the adequacy of the result such that the analyticdeconvolution yields a solution having acceptable error levels;

[0188] if adequate, stop 212;

[0189] if desirable to achieve greater accuracy, input 213 to aniterative deconvolution process.

[0190] As with other embodiments, this process may be used to initiateany conventional iterative nonlinear deconvolution technique to furtherrefine the accurate representations. Note that fewer iterations areneeded because of the accurate starting point provided.

EXAMPLE

[0191] For simplicity, a process is described for one-dimensional (1-D)imagery collection such as produced by the HYDICE sensor. With thesetypes of sensors, a single line of optical detectors/pixels orientedtransverse to the direction of platform travel scans the angular line ofsight electronically in a “whiskbroom” operation. McKeown, D. M. et al.,Fusion of HYDICE Hyperspectral Data with Panchromatic Imagery forCartographic Feature Extraction, IEEE Trans. Geoscience and RemoteSensing, 37, No. 3, pp. 1261-1277, May 1999. The individual 1-D rows arelater combined in the HYDICE imaging process to form a matrix forbuilding 2-D images.

Sample Calculation

[0192] Assume J₀, J₁, and J₂ represent a set of dummy variables and

A(R)=ΣJ _(m) H _(m)(R)=J ₀ H ₀ +J ₁ H ₁ +J ₂ H ₂  (75)

[0193] The Hermite polynomials of orders one, two and three are:

H ₀(R)=1

H ₁(R)=2R  (76)

H ₂(R)=4R ²−2

[0194] By substitution:

B(R)=ΣJ _(m) H _(m)(R)=ΣI _(m) R ^(m)=(J ₀−2J ₂)+(2J ₁)R+(4J ₂)R ²  (77)

[0195] It follows that values for I_(m) are given by:

I ₀=(J ₀−2J ₂)

I ₁=2J ₁  (78)

I ₂=4J ₂

[0196] and conversely, $\begin{matrix}{{J_{0} = {I_{0} + {\frac{1}{2}I_{2}}}}{J_{1} = {\frac{1}{2}I_{1}}}{J_{2} = {\frac{1}{4}I_{2}}}} & (79)\end{matrix}$

[0197] Dealing with Eqn. (73) again: $\begin{matrix}{{^{\frac{- R^{2}}{2}}{\sum\limits_{m = 0}^{2}{D_{m}{H_{m}(R)}}}} = {\left( \frac{a}{2b} \right)^{\frac{1}{2}}^{\frac{- R^{2}}{2}}{\sum\limits_{m = 0}^{2}{{J_{m}(2)}^{\frac{m}{2}}{H_{m}(R)}}}}} & (80) \\{or} & \quad \\{{\sum\limits_{m = 0}^{2}{D_{m}{H_{m}(R)}}} = {C{\sum\limits_{m = 0}^{2}{{J_{m}(2)}^{\frac{m}{2}}{H_{m}(R)}}}}} & (81) \\{\quad {= {\left( \frac{a}{2b} \right)^{\frac{1}{2}}{\sum\limits_{m = 0}^{2}{{J_{m}(2)}^{\frac{m}{2}}{H_{m}(R)}}}}}} & \quad\end{matrix}$

[0198] In expanded form, Eqn. (81) reads: $\begin{matrix}{{{D_{0}{H_{0}(R)}} + {D_{1}{H_{1}(R)}} + {D_{2}{H_{2}(R)}}} = {C\left\lbrack {{J_{0}{H_{0}(R)}} + {{J_{1}(2)}^{\frac{1}{2}}{H_{1}(R)}} + {2J_{2}{H_{2}(R)}}} \right\rbrack}} & (82)\end{matrix}$

[0199] Equating coefficients of like terms involving Hermitepolynomials: $\begin{matrix}{{D_{0} = {C\quad J_{0}}}{D_{1} = {C\quad {J_{1}(2)}^{\frac{1}{2}}}}{D_{2} = {2C\quad J_{2}}}} & (83)\end{matrix}$

[0200] To obtain the solution of the convolution problem, substituteEqns. (78) into Eqns. (82): $\begin{matrix}{{D_{0} = {C\left\lbrack {I_{0} + {\frac{1}{2}I_{2}}} \right\rbrack}}{D_{1} = {{C\left( \frac{1}{2} \right)}^{\frac{1}{2}}I_{1}}}{D_{2} = \left( \frac{C}{2} \right)^{\frac{1}{2}}}} & (84)\end{matrix}$

[0201] To solve the deconvolution problem, these relations are inverted:$\begin{matrix}{{I_{0} = {\left\lbrack \frac{D_{0}}{C} \right\rbrack - \left\lbrack \frac{D_{2}}{C} \right\rbrack}}{I_{1} = {(2)^{\frac{1}{2}}\left\lbrack \frac{D_{1}}{C} \right\rbrack}}{I_{2} = {2\left\lbrack \frac{D_{2}}{C} \right\rbrack}}} & (85)\end{matrix}$

[0202] where, remembering Eqn. (6):

[0203] a is a measure of the relative width, and

[0204] b is an inverse measure of the relative height of the Gaussiansystem PSF and

[0205] D₀, D₁, and D₂ are the full set of three actual (undistorted)image parameter measures.

[0206] While the invention has been described in terms of its preferredembodiments, those skilled in the art will recognize that the inventioncan be practiced with modifications within the spirit and scope of theappended claims. For example, although the system is described inspecific examples for topography, it will operate in areas of medicine,communications, and even business models where deconvolution is apreferred procedure. Thus, it is intended that all matter contained inthe foregoing description or shown in the accompanying drawings shall beinterpreted as illustrative rather than limiting, and the inventionshould be defined only in accordance with the following claims and theirequivalents.

We claim:
 1. A process for deconvolving data collected by a device thepoint spread function response of which follows a Gaussian distribution,said process undertaken to accurately estimate actual parametersderivable from said data, comprising: forming at least one mathematicalrelationship having a first mathematical equivalent of said data on oneside of an equality and a second mathematical equivalent of saidparameters on the other side of said equality; selecting an order, m, ofa Hermite function for modifying said at least one mathematicalrelationship; modifying said mathematical relationship to form at leastone Hermite function therein, wherein forming said at least one Hermitefunction permits identification of at least one like item, having acoefficient, on each side of said equality, said coefficients associatedwith said actual parameters being unknown; developing at least one setof linear equations from said mathematical relationship that relate tosaid coefficient of each said at least one like items; solving said setof linear equations for said unknown coefficients, wherein solving saidset of linear equations produces an exact solution to said convolution;and deconvolving said set of linear equations.
 2. The process of claim 1in which said mathematical relationship is of the form:D(z)=∫p(z−x)I(x)dx where: D(z)=said data; I(x)=expression involving saidparameters; and p(x) point spread function (PSF) representing responsedistribution of said device. and$Y_{m} = {\int_{- \infty}^{\infty}{^{- \frac{{({z - x})}^{2}}{2}}^{- \frac{x^{2}}{2}}{H_{m}(x)}{x}}}$

where: Y_(m) is said data represented at least partially as a Hermitefunction said point spread function for said device is represented by$^{- \frac{z - x^{2}}{2}},$

said Hermite Function is represented by${^{- \frac{x^{2}}{2}}{H_{m}(x)}},$

 and H_(m)(x) is a Hermite polynomial of order m.
 3. The process ofclaim 1 initiating any conventional iterative deconvolution techniquesto further refine said data, wherein fewer iterations will be needed ascompared to conventional methods of deconvolution because of theaccurate starting point provided by said process.
 4. A process fordeconvolving output from detectors, the point spread function responseof said detectors following a Gaussian distribution, said processundertaken to accurately estimate environments derivable from saidoutput, comprising: forming at least one mathematical relationshiphaving a first mathematical equivalent of said output on one side of anequality and a second mathematical equivalent of said environments onthe other side of said equality; selecting an order, m, of a Hermitefunction for modifying said equality, wherein, said order alsodetermines the number of terms for the representation of saidenvironments; modifying said equality to form at least one Hermitefunction within the equality, wherein forming said Hermite functionpermits identification of at least one like item, having a coefficient,on each side of said equality; developing at least one set of linearequations from said equality that relate to said coefficient of eachsaid at least one like items; and solving said set of linear equationsfor said coefficient of each said like item, wherein solving said set oflinear equations produces an exact solution to said convolution, inturn, permitting deconvolution using linear relationships.
 5. Theprocess of claim 4 in which said mathematical relationship is of theform: D(z)=∫p(z−x)I(x)dx where: D(z)=said output; I(x)=saidenvironments; and p(z−x)=a transformed function of the point spreadfunction (PSF) representing response distribution of said device
 6. Theprocess of claim 4 initiating any conventional iterative nonlineardeconvolution technique to further refine said output, wherein feweriterations will be needed as compared to conventional methods ofdeconvolution because of the accurate starting point provided by saidprocess.
 7. A process yielding accurate representations of actual imagedata by deconvolving image data collected by optical detectors, thepoint spread function response of said detectors following a Gaussiandistribution, comprising: establishing a first mathematical relationshipas a general optical convolution integral equating said actual imagedata to said collected image data; selecting a Fourier-Hermite function;employing a generating function for Hermite polynomials for expanding aFourier-Hermite series of said Fourier-Hermite function to establish alinear mathematical relationship between said actual and said collectedimage data; selecting an order, m, of said Fourier-Hermite polynomial,wherein, m also determines the number of terms to be used for therepresentation of said images; expanding said actual image data in saidFourier-Hermite form with unknown coefficients by employing a series ofspecial transformations to convert the side of said mathematicalrelationship representing the actual image to a Fourier-Hermite series;expanding said collected image data with known coefficients in saidFourier-Hermite form; equating said known and unknown coefficients oflike terms on each side of the mathematical relationship to relate thecoefficients of said actual and said collected image data; selecting analgorithm represented by a set of linear equations; solving said linearequations exactly, wherein using said process yields a form proportionalto a Gaussian function times a power series that is defined with afinite number of terms and incorporates said unknown coefficients ofsaid actual image data as presented in a Hermite function, and wherein asolution in closed analytic form provides a satisfactory solution tosaid general definitional convolution integral without approximation;and performing an analytic deconvolution of said convolution equation byinverting said linear equations, wherein said analytic deconvolutionyields a solution having acceptable error levels.
 8. The process ofclaim 7 initiating any conventional iterative deconvolution techniquesto further refine said accurate representations, wherein feweriterations will be needed as compared to conventional methods ofdeconvolution because of the accurate starting point provided by saidprocess.